Optimal. Leaf size=126 \[ -\frac {6 a \sin (c+d x)}{d^4}+\frac {6 a x \cos (c+d x)}{d^3}+\frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {a x^3 \cos (c+d x)}{d}-\frac {24 b \cos (c+d x)}{d^5}-\frac {24 b x \sin (c+d x)}{d^4}+\frac {12 b x^2 \cos (c+d x)}{d^3}+\frac {4 b x^3 \sin (c+d x)}{d^2}-\frac {b x^4 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.31, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {6742, 3296, 2637, 2638} \[ \frac {3 a x^2 \sin (c+d x)}{d^2}-\frac {6 a \sin (c+d x)}{d^4}+\frac {6 a x \cos (c+d x)}{d^3}-\frac {a x^3 \cos (c+d x)}{d}+\frac {4 b x^3 \sin (c+d x)}{d^2}+\frac {12 b x^2 \cos (c+d x)}{d^3}-\frac {24 b x \sin (c+d x)}{d^4}-\frac {24 b \cos (c+d x)}{d^5}-\frac {b x^4 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 6742
Rubi steps
\begin {align*} \int x^3 (a+b x) \sin (c+d x) \, dx &=\int \left (a x^3 \sin (c+d x)+b x^4 \sin (c+d x)\right ) \, dx\\ &=a \int x^3 \sin (c+d x) \, dx+b \int x^4 \sin (c+d x) \, dx\\ &=-\frac {a x^3 \cos (c+d x)}{d}-\frac {b x^4 \cos (c+d x)}{d}+\frac {(3 a) \int x^2 \cos (c+d x) \, dx}{d}+\frac {(4 b) \int x^3 \cos (c+d x) \, dx}{d}\\ &=-\frac {a x^3 \cos (c+d x)}{d}-\frac {b x^4 \cos (c+d x)}{d}+\frac {3 a x^2 \sin (c+d x)}{d^2}+\frac {4 b x^3 \sin (c+d x)}{d^2}-\frac {(6 a) \int x \sin (c+d x) \, dx}{d^2}-\frac {(12 b) \int x^2 \sin (c+d x) \, dx}{d^2}\\ &=\frac {6 a x \cos (c+d x)}{d^3}+\frac {12 b x^2 \cos (c+d x)}{d^3}-\frac {a x^3 \cos (c+d x)}{d}-\frac {b x^4 \cos (c+d x)}{d}+\frac {3 a x^2 \sin (c+d x)}{d^2}+\frac {4 b x^3 \sin (c+d x)}{d^2}-\frac {(6 a) \int \cos (c+d x) \, dx}{d^3}-\frac {(24 b) \int x \cos (c+d x) \, dx}{d^3}\\ &=\frac {6 a x \cos (c+d x)}{d^3}+\frac {12 b x^2 \cos (c+d x)}{d^3}-\frac {a x^3 \cos (c+d x)}{d}-\frac {b x^4 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}-\frac {24 b x \sin (c+d x)}{d^4}+\frac {3 a x^2 \sin (c+d x)}{d^2}+\frac {4 b x^3 \sin (c+d x)}{d^2}+\frac {(24 b) \int \sin (c+d x) \, dx}{d^4}\\ &=-\frac {24 b \cos (c+d x)}{d^5}+\frac {6 a x \cos (c+d x)}{d^3}+\frac {12 b x^2 \cos (c+d x)}{d^3}-\frac {a x^3 \cos (c+d x)}{d}-\frac {b x^4 \cos (c+d x)}{d}-\frac {6 a \sin (c+d x)}{d^4}-\frac {24 b x \sin (c+d x)}{d^4}+\frac {3 a x^2 \sin (c+d x)}{d^2}+\frac {4 b x^3 \sin (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 82, normalized size = 0.65 \[ \frac {d \left (3 a \left (d^2 x^2-2\right )+4 b x \left (d^2 x^2-6\right )\right ) \sin (c+d x)-\left (a d^2 x \left (d^2 x^2-6\right )+b \left (d^4 x^4-12 d^2 x^2+24\right )\right ) \cos (c+d x)}{d^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 85, normalized size = 0.67 \[ -\frac {{\left (b d^{4} x^{4} + a d^{4} x^{3} - 12 \, b d^{2} x^{2} - 6 \, a d^{2} x + 24 \, b\right )} \cos \left (d x + c\right ) - {\left (4 \, b d^{3} x^{3} + 3 \, a d^{3} x^{2} - 24 \, b d x - 6 \, a d\right )} \sin \left (d x + c\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 86, normalized size = 0.68 \[ -\frac {{\left (b d^{4} x^{4} + a d^{4} x^{3} - 12 \, b d^{2} x^{2} - 6 \, a d^{2} x + 24 \, b\right )} \cos \left (d x + c\right )}{d^{5}} + \frac {{\left (4 \, b d^{3} x^{3} + 3 \, a d^{3} x^{2} - 24 \, b d x - 6 \, a d\right )} \sin \left (d x + c\right )}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 359, normalized size = 2.85 \[ \frac {\frac {b \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}+a \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )-\frac {4 b c \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d}-3 a c \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )+\frac {6 b \,c^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d}+3 a \,c^{2} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )-\frac {4 b \,c^{3} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d}+a \,c^{3} \cos \left (d x +c \right )-\frac {b \,c^{4} \cos \left (d x +c \right )}{d}}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 306, normalized size = 2.43 \[ \frac {a c^{3} \cos \left (d x + c\right ) - \frac {b c^{4} \cos \left (d x + c\right )}{d} - 3 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c^{2} + \frac {4 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{3}}{d} + 3 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} a c - \frac {6 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b c^{2}}{d} - {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} a + \frac {4 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b c}{d} - \frac {{\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b}{d}}{d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 122, normalized size = 0.97 \[ \frac {6\,a\,x\,\cos \left (c+d\,x\right )+12\,b\,x^2\,\cos \left (c+d\,x\right )}{d^3}-\frac {6\,a\,\sin \left (c+d\,x\right )+24\,b\,x\,\sin \left (c+d\,x\right )}{d^4}-\frac {a\,x^3\,\cos \left (c+d\,x\right )+b\,x^4\,\cos \left (c+d\,x\right )}{d}+\frac {3\,a\,x^2\,\sin \left (c+d\,x\right )+4\,b\,x^3\,\sin \left (c+d\,x\right )}{d^2}-\frac {24\,b\,\cos \left (c+d\,x\right )}{d^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.34, size = 151, normalized size = 1.20 \[ \begin {cases} - \frac {a x^{3} \cos {\left (c + d x \right )}}{d} + \frac {3 a x^{2} \sin {\left (c + d x \right )}}{d^{2}} + \frac {6 a x \cos {\left (c + d x \right )}}{d^{3}} - \frac {6 a \sin {\left (c + d x \right )}}{d^{4}} - \frac {b x^{4} \cos {\left (c + d x \right )}}{d} + \frac {4 b x^{3} \sin {\left (c + d x \right )}}{d^{2}} + \frac {12 b x^{2} \cos {\left (c + d x \right )}}{d^{3}} - \frac {24 b x \sin {\left (c + d x \right )}}{d^{4}} - \frac {24 b \cos {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\left (\frac {a x^{4}}{4} + \frac {b x^{5}}{5}\right ) \sin {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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